Conclusions and Perspectives

Abstract

In this thesis we discussed many variants of the random (or stochastic) Euclidean matching problems, i.e., matching problems between points randomly generated on a certain Euclidean domain. We supposed the cost of the matching depending on the Euclidean distances of the matched points only. In particular, we tried to evaluate the average optimal cost of the optimal matching. We introduced also the concept of correlation function for the optimal matching, a quantity that, in this optimization problem, is meaningful because of the underlying Euclidean support. We investigated both these quantities (average optimal cost and correlation function) with different methods, inspired by very different research areas of mathematics and physics.